Jaime has been working hard on her tipi. She has decided to use the following pattern as part of the design on the material. She thinks that if she uses cloth triangles, that she can sew them onto the cloth of the tipi to make a pattern. This red, black, red pattern is going to stretch along the outside bottom edge of the tipi.

“That is very cool,” her sister Lily said admiring Jaime’s sketch.

“I think so too. It should look very pretty on the brown material,” Jaime added.

“Yes. How do you know that the triangles will be the exact same size?”

“That’s easy. I can measure the top angle of the triangle using a protractor and then create each one by tracing. This one will have an angle of
. when I am done,” Jaime explained to Lily.

“What kind of triangle is that?” Lily asked.

Do you know? This Concept is all about different types of triangles. As you learn all about the different types of triangles, think about this problem. What type of triangle is in the pattern? Can you justify your answer?

Guidance

In this Concept, we will examine different kinds of triangles. As you know, triangles are geometric figures that have three sides and three angles. There are different types of triangles too. We can classify or identify them in different ways. One way is by their angles measures and one way is by their sides lengths.

Let’s start with angle measures first.

We all know that triangles have three angles. The corners where each of the line segments connects form an angle. When we know the measures of these angles, we can use this information to name and identify the triangles.

Let’s look at some of the different types of triangles according to angle measure.

Acute Triangles
are triangles that have three angles that are less than
. The word “acute” when applied to angles means less than
, so an acute triangle has three angles where all three angles are less than
.

Right Triangles
are triangles with one
angle. The other two angles will be acute, but the key to identifying a right triangle is that it has one right angle.

Obtuse Triangles
are triangles with one angle that is obtuse or greater than
.

Now let's apply this information.

Identify whether each of the following triangles is acute, obtuse or right.

Now let’s break each one down.

With the first triangle, triangle
, we can see that one of the angles is greater than 90 degrees. This is an obtuse triangle.

The second triangle has three angles that are less than 90 degrees, this is an acute triangle.

The third triangle also has three angles that are less than 90 degrees. This is also an acute triangle.

The fourth triangle has a right angle. You can see that because it forms a nice neat corner so perfectly. This is a right triangle.

The fifth triangle has an angle greater than 90 degrees. This is an obtuse triangle.

We can also classify or identify triangles by the length of their sides. This means that we look at the line segments that create the triangle.

Equilateral Triangles
are triangles with all three sides equal.

Isosceles Triangle
has two sides that are equal in length. Often an isosceles triangle is the trickiest one to identify.

Scalene Triangles
are triangles where none of the sides are the same length. All three sides are different lengths.

Now let’s apply what we have learned and identify some triangles.

Classify each triangle as equilateral, isosceles, or scalene.

We need to examine the lengths of the sides in each triangle to see if any sides are congruent.

In the first triangle, two sides are 7 meters long, but the third side is shorter. Which kind of triangle has two congruent sides?
This is an isosceles triangle.

Now let’s look at the second triangle. All three sides are the same length, so this must be an
equilateral triangle.

The last triangle has sides of 5.5 cm, 4.1 cm, and 8 cm. None of the sides are congruent, so this is a
scalene triangle.

Equilateral triangles do not quite fit this pattern. They are always acute. This is because the three angles in an equilateral triangle always measure
.

There is one more thing to know about classifying triangles by their angles and sides. We can also tell whether a triangle is isosceles, scalene, or equilateral by its angles. Every angle is related to the side opposite it. Imagine a book opening. The wider you open it, the greater the distance between the two flaps. In other words, the wider an angle, the longer the side opposite it is. Therefore we can say that if a triangle has two congruent angles, it must have two congruent sides, and thus it must be isosceles. If it has three angles of different measures, then its sides are also all of different lengths, so it is scalene. Finally, an equilateral triangle, as we have seen, always has angles of
, and these angles are opposite congruent sides.

Once you know all of this information, you will find that you can classify a triangle by both its sides and its angles.

Example A

Define a scalene triangle.

Solution: A triangle where all of the side lengths are different.

Example B

Define an obtuse triangle.

Solution: A triangle where one angle is obtuse or greater than
.

Example C

Define an isosceles triangle

Solution: A triangle with two sides of the same length.

Now let's go back to the dilemma from the beginning of the Concept.

What type of triangle is in the pattern? Can you justify your answer?

The triangle in the pattern is an obtuse triangle. It is an obtuse triangle because it has an angle that is greater than
. The other two angles are less than
. Because the largest angle is
, it is an obtuse triangle.

Vocabulary

Acute Triangle

A triangle where all three angles are less than
.

Right Triangle

A triangle with one
angle and two acute angles.

Obtuse Triangle

a triangle with one angle that is greater than
.

Equilateral Triangle

all three side lengths and all three angles are congruent.

Isosceles Triangle

two side lengths are the same.

Scalene Triangle

all three side lengths are different

Congruent

means exactly the same, having the same measure.

Guided Practice

Here is one for you to try on your own.

Identify each triangle by both its sides and angles.

Solution

The first triangle is a right isosceles triangle. It has one right angle and two sides that are the same length.

The second triangle is an acute scalene triangle. All three sides are different lengths and all three angle measures are acute.

The third triangle is an obtuse scalene triangle. It has one obtuse angle and three different side lengths.

The last triangle is an obtuse isosceles. It has one obtuse angle and two side lengths that are the same.